Experimental computation with oscillatory integrals

A previous study by one of the present authors, together with D. Borwein and I. E. Leonard [8], studied the asymptotic behavior of the p-norm of the sine function: sinc(x) = (sinx)/x and along the way looked at closed forms for integer values of p. In this study we address these integrals with the tools of experimental mathematics, namely by computing their numerical values to high precision, both as a challenge in itself, and also in an attempt to recognize the numerical values as closed-form constants. With this approach, we are able to reproduce several of the results of [8] and to find new results, both numeric and analytic, that go beyond the previous study.